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A378294 Nonnegative norms of ideals in Q(sqrt(10), sqrt(26)).

%I A378294 #17 Nov 28 2024 13:29:17
%S A378294 0,1,2,4,5,8,9,10,13,16,18,20,25,26,32,36,37,40,45,49,50,52,64,65,67,
%T A378294 72,74,79,80,81,83,90,98,100,104,117,121,125,128,130,134,144,148,158,
%U A378294 160,162,163,166,169,180,185,191,196,197,199,200,208,225,227,234,242,245,250
%N A378294 Nonnegative norms of ideals in Q(sqrt(10), sqrt(26)).
%C A378294 The number of terms n up to x is asymptotic to c*x/log(x)^(3/4) for an explicitly computable constant c.
%F A378294 A positive integer n is in this sequence if and only if all prime factors of n congruent to {3, 7, 11, 17, 19, 21, 23, 27, 29, 31, 33, 41, 43, 47, 51, 53, 57, 59, 61, 63, 69, 71, 73, 77, 87, 89, 97, 99, 101, 103, 107, 109, 111, 113, 119, 127, 131, 133, 137, 139, 141, 147, 149, 151, 153, 157, 161, 167, 171, 173, 177, 179, 181, 183, 189, 193, 201, 207, 211, 217, 219, 223, 229, 233, 237, 239, 241, 243, 249, 251, 257, 259, 261, 263, 269, 271, 277, 279, 281, 283, 287, 291, 297, 301, 303, 309, 313, 319, 327, 331, 337, 339, 341, 343, 347, 349, 353, 359, 363, 367, 369, 371, 373, 379, 381, 383, 387, 389, 393, 401, 407, 409, 411, 413, 417, 419, 421, 423, 431, 433, 443, 447, 449, 451, 457, 459, 461, 463, 467, 469, 473, 477, 479, 487, 489, 491, 493, 497, 499, 501, 503, 509, 513, 517} mod 520 occur with even exponents.
%e A378294 74=2*37. Using the formula above, 74 is in this sequence.
%e A378294 849=3*283. Using the formula above, 849 is not in this sequence, although 849=1849-1000=43^2-10*100=43^2-10*10^2 and 849==329 (mod 520).
%o A378294 (Magma)
%o A378294 IsRepresentablePrime:=func<p,D | D eq 0 select false else IsSquare(D) select true else KroneckerSymbol(FundamentalDiscriminant(D), p) in [0,1]>;
%o A378294 IsRepresentableMulti:=func<p,S | forall{k: k in Subsets(S) | IsRepresentablePrime(p,&*k)}>;
%o A378294 IsRepresentablePoly:=func<n,S | n eq 0 or (Min(S) gt 0 or n gt 0) and forall{t: t in Factorization(n) | t[2] mod 2 eq 0 or IsRepresentableMulti(t[1],S)}>;
%o A378294 [n: n in [0..250] | IsRepresentablePoly(n,{10, 26})];
%Y A378294 Cf. A378295.
%K A378294 nonn
%O A378294 0,3
%A A378294 _Jovan Radenkovicc_, Nov 22 2024